The Mandelbrot Set

Benoit Mandelbrot extended the work of Fatou and Julia by working on a variation of their formulas.

He was analyzing a series of numbers defined like this

\[z_0=0, z_{n+1} = z_{n}^{2} + c;\; c \in \mathbb{C}\]

and was interested in those options for \(c\), where the series would have an upper limit.

These numbers, fulfilling the following conditions, are called the Mandelbrot Set \(\mathbb{M}\).

\[z_0=0, z_{n+1} = z_{n}^{2} + c;\; c \in \mathbb{M} \Longleftrightarrow \lim_{n \rightarrow \infty} |z_n| \leq g.\]

It’s easy to see that \(g = 2\) here, which will be of importance later on.